Shape optimization problems with competing terms

In this thesis we discuss several shape optimization problems in which the cost functionals are given by the product of two competing terms. For these problems, classical strategies based on the monotonicity of the energy with respect to the set inclusion are not applicable. The definitions and preliminaries required to present our results are collected and discussed in Chapter 1. In Chapter 2 we consider the p-torsional rigidity and the p-first eigenvalue of the Dirichlet p-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. In Chapter 3 we further investigate the above optimization problems, restricting ourselves to the linear case. In Chapter 4 we discuss the behaviors of the mean-to-max ratio of the p-torsion function with respect to the geometry of the domain. In Chapter 5 we study a general version of the classical Cheeger inequality. Finally, in Chapter 6 we look at the optimization problems for the shape functional defined by Jq(Ω) = P(Ω)T^q(Ω) among open sets with prescribed measure